Please explain if it is possible I might develop tetanus after puncture wound on arm from barbed wire fence?

It is possible. Especially if it was dirty, old ; rusty. Is your tetanus status up to date?
PossibleIfNot Immuni. Yes it is possible if you have not been properly immunized against Tetanus and are not current on your booster.That means if it has been more than 10 years since you had a Tetanus booster.So make sure you get a DT or Tdap booster every 10 years as an adult.